This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1

The instability of Wilton ripples

Olga Trichtchenko1†, Bernard Deconinck2 and Jon Wilkening 3

1Department of Mathematics, University College London, Gower Street, London, WC1E 6BT,UK

2Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925,USA

3Department of Mathematics and Lawrence Berkeley National Laboratory, University ofCalifornia, Berkeley, CA 94720-3840, USA

(Received xx; revised xx; accepted xx)

Wilton ripples are a type of periodic traveling wave solution of the full water wave problemincorporating the effects of surface tension. They are characterized by a resonancephenomenon that alters the order at which the resonant harmonic mode enters in aperturbation expansion. We compute such solutions using non-perturbative numericalmethods and investigate their stability by examining the spectrum of the water waveproblem linearized about the resonant traveling wave. Instabilities are observed thatdiffer from any previously found in the context of the water wave problem.

1. Introduction

In 1915 J. R. Wilton (Wilton 1915) included the effects of surface tension and con-structed a series expansion in terms of the amplitude of one-dimensional periodic wavesin water of infinite depth, extending Stokes’s work (Stokes 1847). He noticed that ifthe (non-dimensionalized) coefficient of surface tension equals 1/n (n ∈ Z+), the Stokesexpansions giving travelling wave solutions to Euler’s equations are singular. As a wayto rectify the problem, he modified the form of the perturbation expansion so that thenth harmonic enters at order (n− 1) or (n− 2) instead of n. The resulting solutions areknown as resonant harmonics or Wilton ripples.

The occurrence of Wilton ripples is not merely a mathematical phenomenon. Hendersonand Hammack (Henderson & Hammack 1987) generated and observed such waves in acontrolled tank experiment. In the experiment, several sensors were placed at differentpoints along the length of the tank. They measured the wave profile and the frequenciesof the wave as it travelled down the tank. Even though waves of roughly 20Hz weregenerated by the paddles at one end of the tank, frequencies around 10Hz were observedas well. This is a manifestation of Wilton ripples.

McGoldrick contributed significantly to the understanding of gravity-capillary wavesand their relation to resonant interaction, using both experiment and theory. He demon-strated experimentally that gravity-capillary waves lose their initial profile as theypropagate (McGoldrick 1970a). He also examined these waves using weakly nonlineartheory (McGoldrick 1970b) and used the method of multiple scales (McGoldrick 1971)to investigate the evolution of the gravity-capillary waves. Further, resonant phenomenasuch as Wilton ripples have been studied in model equations. For instance Boyd andHaupt (Haupt & Boyd 1988) investigated Wilton ripples in the context of the so-calledsuper Korteweg-de Vries or Kawahara (Kawahara 1972) equation by adding resonantharmonics into the series expansion, following Wilton’s original approach (Stokes 1847).

† Email address for correspondence: o.trichtchenko@ucl.ac.uk

2 O. Trichtchenko et al

Akers and Gao (Akers & Gao 2012) derived an explicit series solution for the Wiltonripples in this same context.

Not much work has been done analyzing the stability of Wilton ripples. In fact, weare aware only of the work of Jones (Jones 1996, 1992). He investigated a system ofcoupled partial differential equations describing up to cubic order the interaction of thefundamental mode of a gravity-capillary wave with its second harmonic. He also providedwave train solutions of these equations. These were used to examine the stability ofgravity-capillary waves as different parameters are varied. We will analyze the stability ofresonant solutions by looking at the stability eigenvalue problem obtained by linearizingaround a steady state solution. This was previously done by McLean (1982) (McLean1982) who built on numerical work of Longuet-Higgins (Longuet-Higgins 1978a,b) as wellas others to examine growth rates of instabilities as a function of wave steepness. Wewill also use the ideas seen in MacKay and Saffman (1986) (MacKay & Saffman 1986)and use the Hamiltonian structure of the problem in order to find where instabilities canoccur.

In this paper, working with fully nonlinear solutions of the water wave equations, weinvestigate the spectral stability of resonant gravity-capillary waves using the Fourier-Floquet-Hill method (Deconinck & Kutz 2006). To our knowledge, our work presents thefirst study of the different instabilities to which Wilton ripples are susceptible, withoutrestricting the nature of the disturbances. Our paper follows the previous investigationson the instabilities of one-dimensional periodic travelling gravity waves (Deconinck &Oliveras 2011) and of gravity waves in the presence of weak surface tension (Deconinck& Trichtchenko 2014). More details and a more comprehensive investigation of thedifferent types of solutions, their series expansions, and their instabilities will be publishedelsewhere (Trichtchenko et al. 2015).

2. Computing Resonant Gravity-Capillary Waves

One-dimensional gravity-capillary waves are governed by the Euler equations,

φxx + φzz = 0, (x, z) ∈ D, (1a)

φz = 0, z = −h, x ∈ (0, L), (1b)

ηt + ηxφx = φz, z = η(x, t), x ∈ (0, L), (1c)

φt +1

2

(φ2x + φ2

z

)+ gη = σ

ηxx

(1 + η2x)

3/2, z = η(x, t), x ∈ (0, L), (1d)

which incorporate the effects of both gravity and surface tension, where g is the accel-eration due to gravity and σ is the coefficient of surface tension. Here h is the heightof the fluid when at rest, η(x, t) is the elevation of the fluid surface and φ(x, z, t) isthe velocity potential. As was shown in (Vasan & Deconinck 2013), we can add anarbitrary function Cφ(t) (of time but not space) to the Bernoulli condition (1d), whichwe will do for computational purposes below. We focus on solutions on a periodic domainD = {(x, z) | 0 6 x < L,−h < z < η(x, t)} as shown in figure 1. It is clear thatthe parameter space for the travelling-wave solutions of this problem is extensive. Acomprehensive investigation will be presented in (Trichtchenko et al. 2015). In this briefcommunication, we restrict our attention to solutions for which g = 1, the period L = 2πand the water depth h = 0.05. If one employs the criteria of (Benjamin 1967; Benjamin& Feir 1967; Whitham 1967), this puts us in the shallow water regime, quite differentfrom Wilton (Wilton 1915) who worked with h = ∞. However, it should be noted that

The instability of Wilton ripples 3

x

z

z = η(x, t)

Dz = −h

x = Lx = 0

z = 0

Figure 1. The domain on which we solve Euler’s equations.

the above references distinguishing shallow water from deep water do not incorporatesurface tension, and as such their results do not immediately apply.

The regular perturbation expansion (or Stokes expansion) for a 2π-periodic travellingwater wave takes the form

η(x) = ε cosx+

∞∑k=2

εkηk(x), ηk(x) =

k∑j=2

2ηkj cos(jx), (2)

where the Euler equations are reduced using the travelling wave reduction ∂t → −c∂x.Regular perturbation theory (see, for instance, (Vanden-Broeck 2010)) leads to an ex-pression for ηk(x) with a denominator proportional to the left-hand side of

(g + σ)k tanh(h)−(g + k2σ

)tanh(kh) = 0, (k 6= 1). (3)

We refer to (3) as the resonance condition as it indicates that the k-th harmonic isresonant with the base mode. If resonance occurs, i.e. (3) holds for a certain value of k,say k = K, the regular Stokes expansion breaks down, and it is not possible to determineηK(x) in the form (2). Instead, the resonant harmonic arises in the Stokes series at orderεK−1 or εK−2 (Vanden-Broeck 2010; Wilton 1915). It is easy to see that (3) cannot holdwhen σ = 0. In other words, surface tension is a necessary condition for the occurrenceof resonance. Further, (3) holds for at most one value of k > 2 (Trichtchenko et al. 2015).

To compute travelling solutions of (1a-d), we developed a variant of the boundaryintegral method of Wilkening & Yu for the time-periodic problem (Wilkening & Yu2012), modified to take advantage of the travelling wave assumption. Considering onlythe equations (1c-d), which are valid at the surface z = η, and defining a surface velocitypotential q(x, t) = φ(x, η(x, t), t), we have

−cηx = φz − ηxφx := G(η)q, (4a)

−cqx = P

[−cφzηx −

1

2

(φ2x + φ2

z

)− gη + σ

ηxx

(1 + η2x)

3/2

]. (4b)

Here (4b) is obtained from (1d) by using qt = φt + φzηt at the free surface prior torestricting to a travelling frame. Equation (4a) defines the Dirichlet to Neumann operatorG(η). Further, P is the projection operator onto functions of zero mean: Pf(x) = f(x)−1

2π

∫ 2π

0f(x)dx. The introduction of this operator is required since the left-hand side of

(4b) clearly has zero average. This amounts to including Cφ(t) in (1d) to avoid seculargrowth in φ(t) as the wave travels. In addition, in the next step we invert G(η). Workingwith functions of zero average guarantees the existence of a unique inverse.

As written, (4a-b) is a system of two equations for the two unknown surface variablesq(x) and η(x), linked by φ(x, z) through the solution of Laplace’s equation (1a) in thedomain D. We solve the first equation for q(x) using the inverse G(η)−1 of the Dirichlet

4 O. Trichtchenko et al

to Neumann operator (Craig & Sulem 1993):

q = −cG(η)−1ηx,

(φxφz

)=

1

1 + η2x

(1 −ηxηx 1

)(qx−cηx

). (5)

This determines q, φx and φz on the free surface given η. Equation (4b) may then berewritten as R(c, η) = 0, with

R(c, η) := P

[cφx −

1

2φ2x −

1

2φ2z − gη + σ∂x

(ηx

(1 + η2x)

1/2

)], (6)

where we moved cqx inside P [· · · ] and used qx = φx+φzηx. Next we define the objective

function F (c, η) = 14π

∫ 2π

0R(c, η)2 dx, which is minimized (holding the first Fourier mode

of η fixed at the desired amplitude, η1 = ε/2) using the modified Levenberg-Marquardtmethod developed by Wilkening and Yu in (Wilkening & Yu 2012).

Rather than computing the operator G(η) as described in (Wilkening & Yu 2012)and inverting it in (5), we reverse the algorithm to directly compute the Neumann toDirichlet operator. In more detail, G(η)q can be computed by first solving a second-kind Fredholm integral equation

[12 I + K

]µ = q to find the dipole density µ, and then

evaluating G(η)q =[

12H+G

]µ′, where H is the Hilbert transform. Formulas for K and G

are given in (Wilkening & Yu 2012). The modification is to solve[

12H+G

]µ′ = −cηx for

µ′, which is essentially a second-kind Fredholm integral equation due to H2 = −P ; takean antiderivative to find µ; and evaluate q =

[12 I + K

]µ. The improved accuracy comes

from taking an antiderivative instead of a derivative in the middle step. A similar ideawas used by Sethian and Wilkening (Sethian & Wilkening 2004) in the context of linearelasticity to avoid loss of digits when evolving a semigroup whose generator involves twospatial derivatives of a type of Dirichlet-Neumann operator — the inverse operator canbe computed much more accurately.

Figure 2 displays laptop-computed solutions running compiled C++ code implementingthe method sketched above. We use as many Fourier modes as needed to ensure the high-est modes decay to double or quadruple-precision roundoff thresholds. A key differencebetween these numerical results and those for gravity waves with a small coefficient ofsurface tension (Deconinck & Trichtchenko 2014) is that the Fourier modes no longerdecay monotonically. The solutions computed here show a resonance at the K = 10thmode and its higher harmonics. As the amplitude is increased, the modes neighboringthe resonant modes start to grow as well.

3. Stability

We examine the stability of the solutions of the previous section using the Fourier-Floquet-Hill numerical method described in (Deconinck & Kutz 2006). Convergencetheorems for this method are found in (Curtis & Deconinck 2010; Johnson & Zumbrun2012). Denoting one of the travelling solutions computed above by (η0; q0), we considera perturbed solution

η(x, t) = η0(x−ct)+δη1(x−ct)eλt+. . . , q(x, t) = q0(x−ct)+δq1(x−ct)eλt+. . . . (1)

Here (η1; q1) is the spatial part of the perturbation, bounded for all x, including as |x| →∞. Specifically, η1(x) is not required to be periodic with the same period as η0(x). Notethat Re(λ) > 0 implies exponential growth of the perturbed solution, and thus instabilityof η0(x). Substitution of (1) in the governing equations (1a-d) and neglecting terms oforder δ2 yields a linear (but nonlocal) generalized eigenvalue problem for η1, q1 that is

The instability of Wilton ripples 5

Figure 2. Wave profiles for solutions with amplitude ε = 2η1 = 1.244× 10−6, 2.448× 10−6 and4.254×10−6 (top), and semi-log plots of the absolute values of their Fourier modes ηk (bottom).

Here ηk = 12π

∫ 2π

0η(x)e−ikx dx. As expected from the results for gravity waves (Deconinck &

Oliveras 2011), the troughs get wider and the crests become more narrow as the amplitudeincreases. The Wilton ripples also become more apparent, especially in the troughs. For thewave of highest amplitude plotted, a depression is present in the crest.

invariant under spatial translation by 2π; see (Deconinck & Oliveras 2011; Deconinck& Trichtchenko 2014). Therefore we expect η1, q1 to also be eigenfunctions of the shiftoperator, and hence be of Bloch form(

η1(x)q1(x)

)= eiµx

∞∑m=−∞

(NmQm

)eimx =

∞∑m=−∞

(NmQm

)ei(m+µ)x, µ ∈ (−1/2, 1/2]. (2)

Due to the Hamiltonian nature of (1a-d) (Zakharov 1968), the spectrum of this gen-eralized eigenvalue problem is reflection symmetric with respect to both the real andimaginary axes (Wiggins 1990). As a consequence, the presence of any eigenvalue λ offthe imaginary axis implies instability.

For the case of flat water (η0(x) ≡ 0), the spectrum may be computed analytically. Itconsists of all values of the form

λ±µ+m = ic(µ+m)±i√

[g(µ+m) + σ(µ+m)3] tanh((µ+m)h), µ ∈ (−1/2, 1/2], m ∈ Z,(3)

where c =√

(g + σ) tanhh is the wave speed in the linearized regime. Since these valuesare all imaginary, we conclude that flat water is spectrally stable. However, as we examinesolutions with a nonzero amplitude, instabilities arise. Figures 3 and 4 show detailedstability results for the three larger-amplitude solutions of figure 2. Figure 3 shows thecomplex λ plane, while figure 4 shows Re(λ) vs µ. Many phenomena are similar tothose observed for gravity (Deconinck & Oliveras 2011) and (non-resonant) gravity-capillary (Deconinck & Trichtchenko 2014) waves, such as the presence of bubbles ofhigh-frequency instabilities for the larger-amplitude waves. New phenomena are observedas well. We observe nested structures for the two larger-amplitude waves, and, despitebeing in shallow water, we notice the presence of a modulational instability (columns 2and 3). As shown in the right panel of figure 5, the onset of this modulational instability

6 O. Trichtchenko et al

Figure 3. Stability results for the solutions shown in figure 2. The columns correspond toε = 1.244× 10−6, 2.448× 10−6, and 4.254× 10−6, respectively. For each amplitude, two zoomsof the region indicated by the red box above are presented in the second and third rows.

Figure 4. Dependence of Re(λ) on µ for ε = 1.244× 10−6, 2.448× 10−6, and 4.254× 10−6.

occurs around ε = 1.555 × 10−6, when the large bubble of instability present at thatamplitude merges with its mirror image at the origin.

Since eigenvalues are continuous with respect to variations of the wave amplitude(Hislop & Sigal 1996), eigenvalues may leave the imaginary axis as the amplitude increasesonly through collisions on the imaginary axis. This is required to ensure the Hamiltoniansymmetry of the spectrum. Thus, a necessary condition for the loss of stability of η0(x)as the solution bifurcates away from the flat water state is that there exist µ and m suchthat one of the following conditions holds:

λ+µ = λ+

µ+m, λ+µ = λ−µ+m, λ−µ = λ+

µ+m, λ−µ = λ−µ+m. (4)

The instability of Wilton ripples 7

-4e-7

-3e-7

-2e-7

-1e-7

0

1e-7

2e-7

3e-7

4e-7

0 0.05 0.1 0.15 0.2

-8e-11

-4e-11

0

4e-11

8e-11

0.0064975 0.0064986

-2e-6

-1e-6

0

1e-6

2e-6

0 0.1 0.2 0.3 0.4 0.5

Onset of modulational instability

occurs near

Figure 5. (left) Two bubbles of instability nucleate from the origin and move away from theRe(λ) axis in opposite directions as ε increases from 0. The second bubble (with µ < 0) is notshown as the figure is reflection symmetric about µ = 0 (and also about µ = 1/2, by periodicity).(right) At larger amplitudes, the bubble merges with its mirror image to the right, and laterwith its image to the left, at the origin.

For the resonant solutions with K = 10, we have a six-way crossing at λ = 0 when µ = 0,namely

λ+µ = λ−µ = λ+

µ−1 = λ−µ+1 = λ+µ−10 = λ−µ+10 = 0, (µ = 0). (5)

To show that λ+µ−10 = 0 and λ−µ+10 = 0, the resonance condition (3) may be used in

(3). As shown in figure 5, two bubbles of instability nucleate at this six-way crossing.As the wave amplitude ε increases away from zero, these instability bubbles leave theorigin in the Re(λ) vs µ plane in opposite directions, one to the right (shown in figure 5),and the other to the left, a mirror image of the one to the right. For small values ofε, the bubbles are supported on intervals well separated from the origin. Indeed, therange of values µ over which we observe an eigenvalue λ with Re(λ) 6= 0 has the form(−µ1,ε,−µ0,ε) ∪ (µ0,ε, µ1,ε), with 0 < µ0,ε < µ1,ε < 1/2. Although µ0,ε and µ1,ε bothapproach zero as ε→ 0+, the width µ1,ε − µ0,ε of each interval is much smaller than thegap 2µ0,ε between intervals. For example, in the inset of the left panel of figure 5, whenε = 4 × 10−7, the width is 1.09 × 10−6 while the gap is 11900 times larger. Thus, eventhough the instability nucleates at µ = 0, it is not modulational since the wave numbersof the unstable perturbations are tightly confined to a narrow interval separated fromthe origin. In the right panel of figure 5, we see that as ε increases, the bubble grows insize, merges with its reflection about µ = 1/2, and eventually forms a protrusion thatconnects with its reflection about µ = 0 at the origin (around ε = 1.555× 10−6). Beyondthis point, modulational instabilities are present.

We finish these preliminary stability considerations by examining the short-time effectof these instabilities on the water wave profiles they perturb. Given an eigenvalue-eigenfunction pair, the short-time dynamics of the perturbed wave profile is dictatedby the linearized problem obtained above. We have

η(x+ ct, t) ≈ η0(x) + δRe{eiθeλtη1(x)}, η1(x) =

M∑m=−M

Nmei(m+µ)x, (6)

where M is the number of Fourier modes of the computed eigenfunction and θ ∈ (−π, π]is an arbitrary phase; see (Deconinck & Kutz 2006; Deconinck & Oliveras 2011). Sincethe eigenfunction corresponding to λ (associated with −µ) is the complex conjugate of

8 O. Trichtchenko et al

-1.0e-6

0

1.0e-6

2.0e-6

0 60π 120π 180π 240π

-1.0e-6

0

1.0e-6

2.0e-6

0 4π 8π 12π 16π

-1.0e-6

0

1.0e-6

2.0e-6

0 60π 120π 180π 240π

-1.0e-6

0

1.0e-6

2.0e-6

0 4π 8π 12π 16π

-0.004

-0.002

0

0.002

0.004

0 4π 8π 12π 16π

-0.004

-0.002

0

0.002

0.004

0 60π 120π 180π 240π

t=0

t = 250,000 T

t = 1,000,000 T

t=0

t = 250,000 T

t = 1,000,000 T

Figure 6. Three snapshots of a perturbation of the wave in the left column of figure 2 (withamplitude ε = 1.244 × 10−6), approximated by (6) and plotted over 8 (left) or 120 (right)periods of the wave. Here µ = 0.06496517 corresponds to the most unstable eigenvalue, namelyλ = (4.557154 + 2.322777i)×10−7, and T = 2π/c = 28.09599 is the time it takes the underlyingtraveling wave to traverse its wavelength. The unperturbed solution is in the resonant regime,but with secondary oscillations indiscernible since many periods are shown at once.

η1(x), Re(eλtη1(x)) and Re(ieλtη1(x)) span the same space as eλtη1(x) and eλtη1(x). IfRe(λ) 6= 0, Hamiltonian symmetry implies that −λ and −λ are also eigenvalues, andthe eigenfunctions can be obtained by reversing the sign of q (i.e. reversing time) andreflecting space. However, we focus here on linearized solutions that grow as t → +∞rather than decay. The eigenfunctions (η1, q1) are normalized so that

∑|m|6N |Nm|2 = 1,

with complex phase chosen so that N0 is real and positive. The Fourier modes of theeigenfunctions are found to decay exponentially, so it is not difficult to resolve a giveneigenfunction to double-precision accuracy.

Figure 6 shows the results of seeding the traveling solution η0(x − ct) of amplitudeε = 1.244×10−6 with a multiple of the most unstable eigenfunction. This travelling wavecorresponds to the left panels of figures 2, 3 and 4. From the results of figure 4, Re(λ) ismaximized at µ = 0.06496417 by λ = (4.557154 + 2.322777i)× 10−7. The approximation(6) was used with θ = 0 and δ = ε/200. With the above normalization

∑m |Nm|2 = 1,

we have

‖η1‖∞ = max06x62π |η1(x)| = 2.159, ‖δη1‖∞/‖η0‖∞ = 0.00639.

The left column of figure 6 shows eight periods of the traveling wave while the rightcolumn shows 120 periods. In both columns, η(x + ct, t) is plotted, showing the resultsin a frame traveling with the unperturbed wave. The rows show the perturbed solutionat t = 0, t = 250, 000T and t = 1, 000, 000T , where T = 2π/c = 28.09599 is the timerequired for η0(x− ct) to traverse its wavelength. The effect of the initial perturbation isdifficult to discern from η0 in the top row of figure 6. At t = 250, 000T , the perturbationhas grown large enough to be visible in the figure, yielding small ripples in the troughsand regular subharmonic variation in the heights of the wave crests. The third row

The instability of Wilton ripples 9

(t = 1, 000, 000T ) shows the long-time evolution using the linear problem. Unlike thesecond row, these graphs do not represent the nonlinear dynamics of the water wavesurface. Rather, since the perturbation has grown to several orders of magnitude largerthan the profile η0(x), the third panel in effect shows the eigenfunction profile.

These results show that low-amplitude Wilton ripples are remarkably stable. Withε = 1.244 × 10−6, which already deviates substantially from a sinusoidal wave profile(recall figure 2), the seeded wave can travel hundreds of thousands of wavelengths beforelosing coherence. Since Re(λ) decreases rapidly as ε → 0 (recall figure 5), this effect iseven more pronounced at smaller amplitude. The two larger-amplitude waves studiedin detail in figures 2, 3 and 4 are much less stable, with multiple unstable branches ofeigenvalue curves and larger values of Re(λ), though still small compared to 1/T .

4. Conclusion

Using numerical techniques similar to those in (Deconinck & Oliveras 2011) and(Deconinck & Trichtchenko 2014), as well as those introduced in (Wilkening & Yu2012), we compute periodic traveling wave solutions of the full water wave problem(1a-d) including the effects of surface tension. We focus specifically on solutions whosesmall-amplitude limits are fully resonant, the so-called Wilton ripples. We present thefirst computation of the stability spectra of these solutions, providing an overview of thedifferent types of instabilities to which they are susceptible. The resonance conditionallows for a collision of six eigenvalues which was not present in non-resonant gravity-capillary waves. The smaller-amplitude resonant waves are found to be nearly spectrallystable, maintaining coherence while travelling hundreds of thousands of wavelengths.For larger-amplitude resonant waves, new types of instabilities are observed, manifestingthemselves as nested structures and Benjamin-Feir-like instabilities present in shallowwater waves. More comprehensive studies of these solutions and their instabilities willbe presented in (Trichtchenko et al. 2015).

This work was supported in part by the National Science Foundation throughgrant NSF-DMS-1008001 (BD), by the EPSRC under grant EP/J019569/1 and byNSERC (OT), and by the Director, Office of Science, Computational and TechnologyResearch, U.S. Department of Energy under contract number DE-AC02-05CH11231(JW). Any opinions, findings, and conclusions or recommendations expressed in thismaterial are those of the authors and do not necessarily reflect the views of the fundingsources.

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